Using induction on $i,$ prove that $(w^R{)}^t=(w^l{)}^R$ for any string $w$ and all $i0.$

Hints: feel free to use the following Theorem in your proof

Let $u,$vin${}^{**},$ then $(uv{)}^R=v^R{u}^R.$

For the following exercises, give a regular expression that represents that described set.

$2.$ The set of strings over $\{a,b,c\}$ in which all the a$\text{'}$s precede the $b^{\text{'}}s,$ which in turn precede the c$\text{'}$s$.$ It is possible that there are no $a^{\text{'}}s,b\text{'}$s$,$ or c$\text{'}$s$.$

$3.$ The same as Exercise $2$ without the null string.

$4.$ The set of strings over $\{a,b\}$ that contain the substring aa and the substring $bb.$